Resident permits and work visa

Yes you can. What you need to do in order to stay on depends on where you come from. If you’re not an EEA citizen and have been married to an Icelandic citizen, you need to find work, or, if you are already working, you apply for a different type of residence permit and have your employer apply for a work permit for you.
If you were married to a foreign citizen and are unemployed, you need to find work and have the employer apply for a work permit for you and then apply for a different kind of residence permit. If you are already working, all you need to do is to apply for a different type of residence permit (b, c), seeing that you’re no longer eligible for a family reunification permit (e).

If you’re an EEA citizen all you need to do is to show that you have adequate means of financial support to be able to stay on in Iceland. However, if your main/only sustenance is derived from the Social Services, that might affect your residence permit.